Refraction at a Spherical Surface

The bending of light at a single curved (spherical) boundary between two transparent media of refractive indices $n_1$ (the medium containing the object) and $n_2$ (the medium the light enters) is worked out, exactly as for a spherical mirror, using the geometry of a paraxial ray from an on-axis object point striking the surface close to its pole and applying Snell's law in the small-angle approximation ($\sin\theta\approx\theta$). Measuring every distance from the pole $P$ of the spherical surface according to the New Cartesian sign convention, and writing $R$ for the (signed) radius of curvature of the surface, this geometry leads to a single formula relating the object distance $u$, image distance $v$, the two refractive indices and $R$: $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}.$ The sign of $R$ follows the same rule as for any other distance in the convention: it is positive if the centre of curvature of the surface lies on the outgoing (right-hand, transmitted-light) side of the pole, and negative if the centre of curvature lies on the incoming (left-hand, object) side. This single formula for one curved surface is the essential building block for the next section, since a real lens is simply a transparent medium bounded by two such spherical (or, as a limiting case, one spherical and one flat) surfaces; …